452 HITCHCOCK. 



We then find that (250) is identically equal to V(i>p6p, aside from the 

 scalar factor g^uSfii^^^z- 



This completes all possible special forms of the vector (185). The 

 only form which cannot be made factorizable by a term pSbp is the 

 vector (241). 



The only remaining irreducible vector to be considered is (195). 

 If there is at least one axis other than /3i, we can write as (196), which 

 factors like (203) if the vector coefficients are diplanar, that is, unless 

 SjSg/Si^ = 0. If so, we may write (196) as 



i3i(6i.r2.r3 + uxr) + ^sih^XiXz + a.Tia-2), (251) 



where f = ii/3i + ^3(83. To see whether a term pSbp can be added 

 to this vector so that it can be factored into 1'4>P&P, we note that it has 

 jSi for a quintuple axis, and /So for a single axis, and the other axis is 



Therefore V<t)pdp, if it exists, must have /3i for a zero, since three of its 

 axes must be zeros, (not necessarily distinct). We may then set up 

 equations, fifteen in number, as for the vector (241) and the left 

 members will be identical with those for (241). The right members 

 will be the same for equations 2, 5, 7, 8, 9, 11, 13, 14, and 15. The 

 remaining equations become 



1. q\r' - q'r\ = 10. q\r" - q'r'\ + q'\r' - q"r\ = b, 



3. p'lq' - P'q'x = 12. p\q" - p'q'\ + p'\q' - p"q\ = h,-y 



4. p'\q"- p"q'\ = -z 

 6. q'\r"-q"r'\ = « 



The reasoning under case 1 is precisely as before, up to the obtaining 

 of the relations g"i= cq and p"i= cp, since none of the new equa- 

 tions are used so far. Then by 4 and 5 we have 



p"r - pr" = 0, p"q - pq" = 0, 



and since p is not zero it follows that q"r — qr" = 0. But, (from 6), 

 putting for q"i and r"i their values cq and cr, we have 



—c{q"r — qr") = u 



Therefore w = 0, which makes (251) reducible contrary to hypothesis. 

 Hence if r\= 1 we cannot have p different from zero. Hence either 

 r'l or p must be zero. 



